[P3wP3wPanda (OP)]

If one were to design a jet engine similarly to a Dyson bladeless fan, would it work like a turbofan?
The turbofan is more efficient than a turbojet because of many factors, one of which is because it uses the rotation of the turbine to drive a larger blade which operates as a ducted fan. Both the thrust from the jet engine and the thrust from the ducted fan contribute to the overall thrust of the engine, making it more efficient.
The bladeless fan operates on a principle dubbed "air multiplication" because the air forced through the ring entrains the air around it, moving many times more air than what is pushed through the tower base. If you replaced the air moving through the device with some kind of jet propulsion, i expect it would have a similar effect with the surrounding air, but would that moving air create extra thrust? Would such an engine be more efficient because of the "air multiplication" effect or is the extra thrust created by the turbofan only because the air is physically pushing against the blade of the fan?

[Bored chemist]

No.
You move more air, but it moves more slowly.
So there's no more thrust overall.